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A Note on Quaternionic Hyperbolic Ideal Triangle Groups

  Published:2016-02-16
 Printed: Jun 2016
  • Wensheng Cao,
    School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, P.R. China
  • Xiaolin Huang,
    School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, P.R. China
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Abstract

In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.
Keywords: quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
MSC Classifications: 20F67, 22E40, 30F40 show english descriptions Hyperbolic groups and nonpositively curved groups
Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Kleinian groups [See also 20H10]
20F67 - Hyperbolic groups and nonpositively curved groups
22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
30F40 - Kleinian groups [See also 20H10]
 

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